{"id":"41e54c09-4221-4a5d-a602-588b7dfa13d1","arxiv_id":"2607.11440","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Minimisers (if any) of the fixed-N Lieb–Oxford bound are necessarily compactly supported for every N≥1.","lead":"Any density that minimises the fixed-N Lieb–Oxford constant must be compactly supported. The result removes a possible obstruction to computing or approximating those constants and extends the classical N=1 case of Lieb and Oxford.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The paper's central claim is a pure non-existence statement for non-compactly-supported minimisers of the fixed-N Lieb–Oxford functional. The proof proceeds by contradiction: any such minimiser yields a Kantorovich potential φ_ρ via the first-order condition (17) and Lemma 1; Theorem 2 upgrades it to a continuous c-conjugated potential; the inner/outer estimates (19)–(20) plus HLS bootstrap give continuity of U_ρ and of ρ itself, vanishing on ∂supp(ρ); the same estimates force the constant c to equal the limit C_φ at infinity; the asymptotic (14) then produces an immediate contradiction both when the complement is unbounded and when a connected component of the support is unbounded. All steps are elementary once Lemma 2 is granted, and Lemma 2 is precisely the result proved in the cited reference under the hypotheses used here. No circularity, no unstated regularity, and no claim of existence. The reader's identification of the asymptotic as the sole external input is accurate; that input is already established, so the verdict remains ACCEPT.","tokens_in":12788,"tokens_out":522,"duration_ms":5408,"concrete_test":"Independently re-derive the asymptotic expansion (14) of Lemma 2 for a c-conjugated KP under the sole hypothesis that supp(ρ) is unbounded (no connectedness), following the argument of [8, Rem. 4]; if the expansion fails for some ρ∈L^{1}∩L^{1+s/d}, the contradiction step on p. 10 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption flag (asymptotics of c-conjugated Kantorovich potentials for every unbounded-support density) is the only plausible soft spot, but it is already settled by the cited result [8, Thm. 1 + Rem. 4] and is used only under the standing hypotheses of Theorem 1. The remainder of the argument (variational characterisation of KPs via Lemma 1, bootstrap of local-to-global integrability via HLS + the inner estimate, and the two-sided contradiction for unbounded support) is self-contained and does not introduce further hidden assumptions. Existence of minimisers is correctly left open.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves that any minimiser (if it exists) of the fixed-N Lieb–Oxford constant c_LO(s,d,N) must be compactly supported. For ρ in L^{1} ∩ L^{1+s/d} with mass N that attains the bound, the authors derive a variational inequality showing that ϕ_ρ := U_ρ − β_s c_LO ρ^{s/d} is a Kantorovich potential for the multimarginal Riesz OT problem. After replacing it by a c-conjugated Lipschitz representative ϕ (via a strong duality formulation (KD’) proved in the appendix), they obtain inner/outer estimates relating U_ρ, ϕ and ρ. A bootstrap that combines these estimates with the Hardy–Littlewood–Sobolev inequality upgrades local integrability of ρ to global L^q integrability for some q > d/(d−s), which forces U_ρ → 0 at infinity and identifies the additive constant of ϕ with its limit at infinity. Assuming unbounded support then yields a two-sided contradiction: the complement of the support cannot be unbounded (by the asymptotic expansion of ϕ), while a bounded complement forces a non-integrable tail of ρ. The argument extends the classical N=1 result of Lieb–Oxford and leaves existence of minimisers open.","tokens_in":12967,"tokens_out":816,"duration_ms":7257,"significance":"The result settles a natural structural question for the fixed-N Lieb–Oxford problem that had been known only for N=1 and only heuristically for N>1. Compact support of putative minimisers is a useful a-priori restriction for any future existence or numerical search, and the proof technique (variational characterisation of Kantorovich potentials + HLS bootstrap + asymptotic comparison) is clean and reusable. The appendix supplies a self-contained justification of the essential-infimum duality (KD’) that is needed because the candidate potential arises only almost everywhere. The paper correctly treats existence as open and relies on a recent, carefully cited asymptotic lemma for c-conjugated potentials; under those hypotheses the argument is complete.","major_comments":[],"minor_comments":[{"comment":"Page 1 and abstract: the arXiv identifier and date appear as 2607.11440 / July 14, 2026; these look like placeholders and should be corrected before publication.","section":null},{"comment":"Section 2, after (KD’): the set I_ρ is defined as L^{1}_loc ∩ L^{1}(ρ); a short remark that this is the natural space for the variational inequality (5) would help the reader who is used to continuous dual potentials.","section":null},{"comment":"Lemma 3 and the subsequent global-integrability bootstrap: the iteration q ↦ sq/(d−(d−s)q) is clear, but an explicit sentence that the same map works globally once ϕ−c ≥ 0 would make the transition from local to global more transparent.","section":null},{"comment":"Appendix A.1: the successive construction of the ϕ_k is correct but a little dense; a one-line indication that each step only uses Fubini and the definition of essential infimum would improve readability.","section":null},{"comment":"References: a few entries (e.g. [7], [8]) are listed as “to appear” or with incomplete pagination; final bibliographic data should be supplied if available.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, technically solid and of clear interest to the mathematical DFT / multimarginal OT community. I see no reason to delay acceptance; the minor presentation points can be handled at the proof stage. The reliance on the recent asymptotic result [8] is properly acknowledged and does not affect the novelty of the compactness argument itself."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles a natural structural question: any fixed-N minimiser of the Lieb–Oxford functional (if one exists) must be compactly supported. That was known only for N=1 since 1981; the extension to arbitrary finite N is the actual new result.\n\nWhat they do well is keep the argument tight. They work with a slightly relaxed Kantorovich dual (KD’), characterise its maximisers variationally (Lemma 1), bootstrap local-to-global integrability of ρ via HLS plus the inner estimate, and then run a two-sided contradiction: unbounded support forces either the complement or a connected component at infinity to violate integrability once the asymptotic expansion of the c-conjugated potential is fed in. The appendix that justifies the strong duality is careful and self-contained. They never claim existence, which remains open, so there is no over-reach.\n\nThe only soft spot is the reliance on the asymptotic expansion of Kantorovich potentials (their Lemma 2, taken from Lelotte 2025). That result is used as a black box, but under the standing hypotheses of Theorem 1 it applies, and the stress-test note is right that nothing else is hidden. Circularity is zero; the argument is pure contradiction. Citations are appropriate and the math is reproducible from the text once the external asymptotic lemma is granted.\n\nThis is for people working on universal functionals, multimarginal OT with Riesz costs, or mathematical DFT. It removes an obstruction for both numerical searches and further analytic work at fixed N. I would send it to a serious referee without hesitation; the result is solid and the write-up is clear enough that a referee can check it in one sitting.","headline":"Clean extension of Lieb–Oxford compactness from N=1 to all finite N; proof holds and existence is correctly left open.","tokens_in":13477,"tokens_out":432,"would_cite":true,"duration_ms":4560,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","35Q40","49J45"],"pacs":[],"model":"grok-4.5","headline":"Minimisers of the fixed-N Lieb–Oxford bound, if they exist, must be compactly supported.","keywords":["Lieb-Oxford inequality","multimarginal optimal transport","Kantorovich potential","Riesz cost","compact support","fixed particle number"],"falsifier":"Exhibit a density of total mass N that is unbounded in support, belongs to L^{1} igcap L^{1+s/d}, and attains the sharp constant c_LO(s,d,N); alternatively, produce a c-conjugated Kantorovich potential whose far-field expansion violates the (N-1)/|x|^s law on an unbounded support.","tokens_in":13744,"feed_emoji":"⚖️","tokens_out":615,"duration_ms":5753,"temperature":0.7,"pith_summary":"The Lieb–Oxford inequality supplies a universal lower bound on the indirect Riesz energy of a system of charged particles. For a fixed finite number of particles N the paper asks whether a density that attains the best possible constant can fail to have compact support. The answer is no: any minimiser must be compactly supported. The argument proceeds by treating the indirect energy as a multimarginal optimal-transport problem, extracting a Kantorovich potential from the first-order optimality condition, and comparing its known decay at infinity with the Riesz potential generated by the density. The resulting contradiction forces the support to be bounded. The result extends the classical one-particle observation of Lieb and Oxford and supplies a structural constraint that any future existence proof or numerical search for the sharp constant must respect.","feed_headline":"Fixed-N Lieb–Oxford minimisers must have compact support","feed_subtitle":"Any density attaining the sharp constant for finite particle number cannot spread to infinity.","key_machinery":"The asymptotic expansion of c-conjugated Kantorovich potentials: any such potential associated with a density of unbounded support behaves at infinity like (N-1)/|x|^s plus a constant. Combined with the variational characterisation that identifies U_ρ - eta_s c_LO \rho^{s/d} as a Kantorovich potential, this expansion produces incompatible decay rates once the support is assumed unbounded.","core_discovery":"If a density ρ of total mass N attains the sharp constant c_LO(s,d,N) in the fixed-N Lieb–Oxford inequality, then ρ is necessarily compactly supported. The claim is proved for every dimension d and every Riesz exponent 0 < s < d, and it does not assert existence of such a minimiser.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Fixed-N LO minimisers cannot have non-compact support","Any density attaining finite-N LO constant is compactly supported","No non-compactly supported minimisers for fixed-N Lieb–Oxford","Finite-particle LO bound forces compact support if attained","Lieb–Oxford fixed-N minimisers (if any) must vanish at infinity"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The proof relies on the known far-field expansion of every c-conjugated Kantorovich potential for densities of unbounded support; if that expansion fails for some admissible density the contradiction argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Fixed-N LO minimisers cannot have non-compact support","Any density attaining finite-N LO constant is compactly supported","No non-compactly supported minimisers for fixed-N Lieb–Oxford","Finite-particle LO bound forces compact support if attained","Lieb–Oxford fixed-N minimisers (if any) must vanish at infinity"]},"model":"grok-4.5","effort":"low","cost_usd":0.004542,"raw_usage":{"total_tokens":1173,"prompt_tokens":576,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":45420000,"prompt_tokens_details":{"text_tokens":576,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":504,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":576,"tokens_out":93,"duration_ms":4763,"temperature":1.0,"reasoning_tokens":504,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T05:34:36.724798+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a density of total mass N that is unbounded in support, belongs to L^{1} igcap L^{1+s/d}, and attains the sharp constant c_LO(s,d,N); alternatively, produce a c-conjugated Kantorovich potential whose far-field expansion violates the (N-1)/|x|^s law on an unbounded support.","supporting_citations":[],"review_version":1}